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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS

Int. J. Numer. Meth. Fluids 2000; 00:1–6 Prepared using fldauth.cls [Version: 2002/09/18 v1.01]

Kernel-Based Vector Field Reconstructionin Computational Fluid Dynamic Models

L. Bonaventura∗♭ , A. Iske♯, E. Miglio♭

♭ MOX – Modellistica e Calcolo ScientificoDipartimento di Matematica “F. Brioschi”

Politecnico di MilanoVia Bonardi 9, 20133 Milano, Italy

♯ Department of MathematicsUniversity of Hamburg

Bundesstraße 55D-20146 Hamburg, Germany

SUMMARY

Kernel-based reconstruction methods are applied to obtain highly accurate approximations oflocal vector fields from normal components assigned at the edges of a computational mesh. Thetheoretical background of kernel-based reconstructions for vector-valued functions is first reviewed,before the reconstruction method is adapted to the specific requirements of relevant applications incomputational fluid dynamics. To this end, important computational aspects concerning the design ofthe reconstruction scheme, such as the selection of suitable stencils, are explained in detail. Extensivenumerical examples and comparisons concerning hydrodynamic models show that the proposed kernel-based reconstruction improves the accuracy of standard finite element discretizations, includingRaviart-Thomas (RT) elements, quite significantly, while retaining discrete conservation propertiesof important physical quantities, such as mass, vorticity or potential enstrophy. Copyright c© 2000John Wiley & Sons, Ltd.

key words:

Kernel-based approximation methods, local vector field reconstruction, Raviart-Thomas (RT)elements, computational fluid dynamics

1. INTRODUCTION

Kernel-based reconstruction methods have recently gained enormous popularity in relevantapplications of computational science and engineering, such as meshfree discretizations forpartial differential equations [11, 12, 13, 14], particle-based numerical simulations [18], machinelearning [36] and manifold learning [21], as well as in many other challenging problems of high-dimensional approximation.

∗Correspondence to: MOX – Modellistica e Calcolo Scientifico, Dipartimento di Matematica “F. Brioschi”,Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

Received

Copyright c© 2000 John Wiley & Sons, Ltd. Revised

2 L. BONAVENTURA, A. ISKE, E. MIGLIO

Radial (reproducing) kernels, also referred to as radial basis functions (RBFs), are powerfultools for interpolation and approximation of scalar-valued multivariate functions. In fact, radialkernels provide highly accurate reconstructions from discrete scattered data without imposingtoo severe restrictions on the spatial distribution of the sample points. Moreover, customizedpreconditioners lead to stable implementations of the reconstruction scheme, whose evaluationis fast – due to the kernel’s simple representation. Therefore, radial kernels are especially verypopular in applications of high-dimensional approximation. For a recent account on theoreticaland practical aspects of radial (reproducing) kernels and their applications, we refer to thetextbooks [7, 17, 40].

Suitable choices for radial kernels include polyharmonic splines, Gaussians, and (inverse)multiquadrics, where the latter two kernels may (for band-limited target functions) lead tospectral convergence rates of the respective interpolation methods. Moreover, it can be shownthat, in this case, also any derivative of the interpolated target function can be approximatedaccurately by the corresponding derivative of the kernel-based interpolant, see [40, Chapter 11]for theoretical details on error bounds.

Quite recently, different concepts of meshfree methods were developed to solve numericallypartial differential equations by radial kernels, where the utilized approaches include collocationmethods [9, 10], Galerkin methods [39], and methods of backward characteristics [4], to mentionbut a few.

But the utility of radial kernels has also been shown for mesh-based methods, such as forinstance in [19, 38] where radial kernels were used to obtain highly accurate finite volumeENO schemes by local reconstruction from scattered cell average values. Radial kernels werealso applied to semi-Lagrangian discretizations in [3], and similar interpolation approachesbased on kriging were used in [34]. In [32, 33], interpolation by radial kernels is essential toachieve accurate semi-Lagrangian schemes on Cartesian grids with cut boundary cells. Anotherexample is the recent adaptive ADER scheme [20], where kernel-based interpolation is used toconstruct appropriate error estimators for mesh adaption. Moreover, radial kernels were usedin [42] to develop high order approximation schemes for discretizing differential operators ofthe shallow water equations.

In all these applications, scalar-valued radial kernels were used for local interpolation,which allows one to reconstruct a function at any point in space, given its scalar values ina neighbourhood of that point. In this paper, we aim at pursuing further this development byusing radial kernels for the interpolation of vector-valued functions. In this case, due to thereconstruction scheme, the radial kernel is required to be matrix-valued rather than scalar-valued.

The overall framework of Hermite-Birkhoff interpolation via matrix-valued radial kernelsis covered in outmost generality through the seminal paper [24] of Narcowich and Ward.Although we believe that the 1994 paper [24] has remarkably great potential for applications incomputational fluid dynamics and related fields, it seems that it has not gained much attentionin applications since then. Indeed, more ad hoc reconstruction approaches were developedinstead, e.g. in [27, 37] for similar problems.

In the present paper, relevant theoretical details from [24] are briefly reviewed first, beforesome of the theory is adapted to the particular requirements of specific applications incomputational fluid dynamics which we wish to address here. To this end, we will specializethe very general method of [24] to design much simpler, yet powerful, reconstruction schemesby diagonally-scaled radial kernels, in which case the matrix-valued radial kernel is given

Copyright c© 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2000; 00:1–6

Prepared using fldauth.cls

KERNEL-BASED VECTOR FIELD RECONSTRUCTION 3

by a diagonal matrix. This way we wish to make the results of [24] more accessible. Theprimary goal of this paper, however, is to provide accurate vector field reconstructions inorder to improve standard finite element discretizations, such as low order Raviart-Thomas(RT) elements [28, 30].

In RT finite element methods, discrete vector fields are usually represented by theircomponents normal to the edges of the computational mesh. Relevant applications of RTelements include electromagnetic [15] and hydrodynamical problems, the latter being the focusof the present paper.

The lowest order RT elements lead easily to numerical methods that exhibit appealingmimetic properties, such as conservation of mass, vorticity and potential enstrophy. Recentmethods for computational fluid dynamics applications, such as [5, 6, 8, 23, 27], rely on theabove mentioned conservation properties. In contrast, although higher order RT elements areavailable (see e.g. [25]), to the best of our knowledge we are not aware of any work on higherorder methods (using RT or other elements) displaying such discrete preservation properties.The kernel-based Gaussian reconstruction method proposed in this paper allows us to enhancethe accuracy of low order RT elements, while retaining the important discrete conservationproperties of the above mentioned schemes.

The outline of this paper is as follows. In Section 2, key features of kernel-based vectorfield reconstruction (by using diagonally-scaled radial kernels) are first reviewed, before thespecific vector reconstruction problem is explained in Section 3. Practical aspects concerningthe implementation of the proposed reconstruction method are also addressed, especially thechoice of suitable stencils. In Section 4, the accuracy of the utilized kernel-based Gaussianreconstruction scheme is assessed in comparison with RT elements of order zero, RT0. This isdone through specific numerical experiments, where accuracy rates are determined numericallyfor both methods. Finally, in Section 5, the practical relevance of the proposed kernel-basedvector field reconstruction method is demonstrated by using two different shallow water models,aiming at atmospheric and coastal modelling, respectively.

2. VECTOR FIELD RECONSTRUCTION FROM HERMITE-BIRKHOFF DATA

This section explains kernel-based reconstruction for vector-valued functions from scatteredHermite-Birkhoff data. In order to discuss this problem, let u : R

d → Rn denote a vector-

valued function, u = (u1, . . . , un). Moreover, assume that for a finite set Λ = {λ}λ∈Λ oflinearly independent vector-valued linear functionals, samples λ(u) ∈ R are given, where theaction of any λ = (λ1, . . . , λn) ∈ Λ on u is defined as

λ(u) =n

∑

k=1

λk(uk).

Specific examples for functionals λ, as we utilize them in relevant hydrodynamical applications,are given in Section 3.

Reconstruction of u from values {λ(u) :λ ∈ Λ} requires finding a suitable recovery functions : R

d → Rn satisfying

λ(s) = λ(u), for all λ ∈ Λ, (1)

or, s∣

∣

Λ= u

∣

∣

Λ, in short hand notation. For the sake of computational efficiency, the approach




taken in this paper uses diagonally-scaled matrix-valued radial kernel functions. In this case,the matrix-valued radial kernel is given by a diagonal matrix.

In order to explain this particular reconstruction scheme more precisely, let Φ : Rd → R

n×n

be a diagonal matrix-valued function,

Φ(x) = diag(φ1(x), . . . , φn(x)) =

φ1(x). . .

φn(x)

∈ R

n×n for x ∈ Rd (2)

with scalar-valued diagonal components φj : Rd → R, 1 ≤ j ≤ n. Moreover, we assume that

Φ is even, Φ(x) = Φ(−x) for all x ∈ Rd, i.e., φj(x) = φj(−x) for all 1 ≤ j ≤ n. Suitable

entries for the diagonal components φj of Φ in (2) are radial kernel functions, φ(r) ≡ φj(r),

r = ‖x‖, where popular choices include Gaussians, φ(r) = e−r2

, and inverse multiquadrics,φ(r) = 1/

√1 + r2.

Now according to the proposed reconstruction scheme, the interpolant s in (1) is has theform

s = τ ∗ Φ with τ =∑

µ∈Λ

cµµ, (3)

where the convolution productτ ∗ Φ : R

d → Rn

between the diagonal matrix Φ and the functional τ = (τ1, . . . , τn) is defined componentwiseas

[τ ∗ Φ]j(x) = τjφj(x − y) for 1 ≤ j ≤ n.

Therefore, any component sj , 1 ≤ j ≤ n, of the reconstruction s = (s1, . . . , sn) in (3) canbe expressed as

sj(x) = [τ ∗ Φ]j(x) =∑

µ∈Λ

cµ[µ ∗ Φ]j(x) =∑

µ∈Λ

cµµyj φj(x − y), (4)

where µyj denotes action of the functional µj on variable y ∈ R

d.Now solving the reconstruction problem (1) under the assumption s = τ ∗ Φ for s in (3)

amounts to solving the N -by-N linear equation system

Ac = u∣

∣

Λ(5)

with unknown c = (cµ)µ∈Λ ∈ RN , where

A = (λ(µ ∗ Φ))µ,λ∈Λ∈ R

N×N , (6)

so that any component aµ,λ = λ(µ ∗ Φ) of A has the form

aµ,λ =

n∑

j=1

λxj µy

j φj(x − y) for µ, λ ∈ Λ.

Note that the linear system (5) has, for any input data, a unique solution, provided that Φis positive definite, which in other words means that the matrix A in (6) is for any combinationof admissible input data – functionals Λ and function values u

∣

∣

Λ– positive definite.




Definition 1. Let Φ : Rd → R

n×n be an even matrix-valued function. We say that Φ ispositive definite, iff the quadratic form

∑

µ,λ∈Λ

cµcλλ(µ ∗ Φ) (7)

is positive for any set Λ = {λ}λ∈Λ of linearly independent functionals and any non-zero vectorc = (cµ) ∈ R

N \ {0}.The class of positive definite matrix-valued functions will be denoted in the following by

PD. It is straightforward to show that any diagonal matrix-valued function Φ is positivedefinite, if and only if all its diagonal components φj , 1 ≤ j ≤ n, are positive definite. Weprove this simple but quite useful characterization later in this section through Lemma ??. Tothis end, let us first make some further preparations. Note that the Definition 1 for positivedefinite vector-valued functions covers in particular the special case of scalar-valued functions,where n = 1. Therefore, it makes sense to require φj ∈ PD for the individual scalar-valuedcomponents of Φ in (2).

But scalar-valued positive definite functions are well-understood. Moreover, the classof radial positive definite functions can be characterized through completely monotonefunctions [22]. This particular characterization – due to I.J. Schoenberg in 1938 – gives a usefulnecessary and sufficient criterion for the construction of radial positive definite kernel functions.For details, we refer to [22]. Important examples for scalar-valued radial positive definite

functions, radial kernels, φ ∈ PD, are Gaussians, φ(r) = e−r2

, and inverse multiquadrics,φ(r) = 1/

√1 + r2, as already mentioned at the outset of this section.

We further remark that the class of positive definite functions is scale-invariant, so that inparticular both φ(r) = e−αr2 ∈ PD and φ(r) = 1/

√α2 + r2 ∈ PD are positive definite for any

scalar α > 0. In fact, this observation is of crucial importance in the numerical experiments ofSections 4 and 5.

Another important observation is concerning the well-posedness of the utilizedreconstruction scheme: if all n scalar-valued diagonal components φj , 1 ≤ j ≤ n, ofΦ = diag(φ1, . . . , φn) are positive definite, then Φ is positive definite. Indeed, this is duethe representation

∑

µ,λ∈Λ

cµcλλ(µ ∗ Φ) =∑

µ,λ∈Λ

cµcλ

n∑

j=1

λxj (µy

j φj(x − y))

=n

∑

j=1

∑

µ,λ∈Λ

cµcλλxj (µy

j φj(x − y))

(8)

for the quadratic form in (7), which is in this case positive for any c ∈ RN \ {0}, i.e., Φ is

positive definite. Note that the converse is also true, i.e., if Φ = diag(φ1, . . . , φn) is positivedefinite, then all φj , 1 ≤ j ≤ n, must be positive definite.

Let us finally remark that the reconstruction scheme of this section can be generalized frompositive definite Φ ∈ PD to conditionally positive definite Φ, by following along the linesof Narcowich and Ward [24]. This would lead to a larger family of potentially available φj ,albeit with a more complicated reconstruction scheme. We omit further technical aspects,since these lengthy details are clearly beyond the aims and scope of this paper. For the sake ofcomputational efficiency, we rather prefer to work with positive definite radial kernels Φ ∈ PD.




Let us conclude the discussion of this section as follows.

Theorem 1. Let Φ = diag(φ1, . . . , φn) be positive definite, Φ ∈ PD, i.e. φj ∈ PD for all1 ≤ j ≤ n. Then the reconstruction problem (1) has a unique solution s of the form

s(x) =∑

µ∈Λ

cµ(µ ∗ Φ)(x),

where the coefficients c = (cµ)µ∈Λ of s can be computed by solving the linear system Ac = u∣

∣

Λ,

whose coefficient matrix A in (6) is positive definite. Any component sj of s = (s1, . . . , sn)can be expressed as

sj(x) =∑

µ∈Λ

cµµyj φj(x − y) for 1 ≤ j ≤ n.

Concrete and relevant examples for s, Φ and Λ are given in the following two sections.

3. LOCAL VECTOR FIELD RECONSTRUCTION IN TWO DIMENSIONS

We now formulate the specific vector reconstruction problem that we wish to address inhydrodynamical applications, some of which are discussed in Section 5. In this particular casewe work with planar vector fields u : R

2 → R2, u = [u1, u2], so that d = 2 and n = 2. Moreover,

we assume we are given a (possibly scattered) set X = {x1, . . . ,xN} ⊂ R2 of N pairwise distinct

planar points, where each xi ∈ X corresponds to a unit vector nTi = [n1

i , n2i ] ∈ R

2, satisfying‖ni‖ = 1, for 1 ≤ i ≤ N .

Now we wish to reconstruct a smooth vector field u : R2 → R

2, from given scalar samplesui = nT

i · u(xi) ∈ R, for 1 ≤ i ≤ N . According to the reconstruction scheme proposed in theprevious section, we are concerned with solving

λi(u) = λi(s), for 1 ≤ i ≤ N,

where λi = nTi · δxi

, and where δxidenotes the Dirac point evaluation functional, defined as

δxi(u) = u(xi) ∈ R

2, for 1 ≤ i ≤ N .Recall that the recovery function s = (s1, s2) : R

2 → R2 is required to have the form (3),

and whose two components can be expressed as in (4). Moreover, we assume

Φ = diag(φ1, φ2) =

[

φ1 00 φ2

]

.

Hence, λi ∗ Φ, 1 ≤ i ≤ N , has the form

(λi ∗ Φ)(x) =

[

φ1(x − xi)n1i

φ2(x − xi)n2i

]

, for 1 ≤ i ≤ N, (9)

and therefore, the reconstruction s in (3) can be expressed as

s(x) =

N∑

i=1

ci

[

φ1(x − xi)n1i

φ2(x − xi)n2i

]

.




Moreover, the unknown coefficients c = (c1, . . . , cN )T ∈ RN of s are computed by solving the

linear system Ac = u∣

∣

Λin (5), where

u∣

∣

Λ= (λ1(u), . . . , λN (u))T = (nT

1 u(x1), . . . ,nTNu(xN ))T ∈ R

N ,

and where the entries in the positive definite matrix A = (aij)1≤i,j≤N ∈ RN×N are given by

aij = λj(λi ∗ Φ) = n1

jn1

i φ1(xj − xi) + n2

jn2

i φ2(xj − xi), for 1 ≤ i, j ≤ N.

4. COMPARISONS BETWEEN GAUSSIAN KERNELS AND RT ELEMENTS

In this section, we present selected numerical experiments in order to assess the effectiveaccuracy order of kernel-based reconstruction by using the positive definite Gaussians, wherewe let φ1(r) ≡ φ2(r) ≡ φ(r) = e−r2α ∈ PD. According to available theoretical bounds [40,Chapter 11], spectral convergence rates can be proven for a very small class of smooth functions.But for hydrodynamic models from realistic application scenarios (in particular for the shallowwater models in Section 5) we rely on less smoothness, so that the theoretical error bounds dono longer apply.

The scaling parameter α in the radial kernel. was chosen in general according to thesuggestion in [32], e.g. by assigning to it a value of the same order of magnitude as thesize of the computational domain. This choice only relies on heuristic arguments and not on arigorous analysis. On the other hand, it has proven to be effective in a number of applications,see e.g. [32, 33]. We have checked that changing the scale value to others of the same orderof magnitude did not apparently affect the results. This choice is however a key point in theapplication of radial kernels and should be investigated in greater depth.

Moreover, we remark that the spectral condition number κ(A) of the arising interpolationmatrix A is large, if the minimal distance between different sample points is small. To furtherexplain this, if two sample points are (nearly) coincident, then two rows (columns) of thematrix A are (nearly) coincident. In this case, the conditioning of the corresponding linearsystem is from a numerical viewpoint very critical – due to a large condition number κ(A).

In order to achieve a reasonable trade-off between the numerical stability of the proposedreconstruction scheme, on the one hand, and its resulting accuracy, on the other hand, we workwith sets of interpolation points, stencils, lying on a hexagonal grid, as shown in Figure 1. In thiscase, the interpolation points of the stencils are well-separated, i.e., small distances between theinterpolation points are avoided. Our numerical results concerning Gaussian reconstruction,as reflected by Tables I and II, are involving four different stencils of sizes N = 3, 9, 15, 21,see Figure 1. For further motivation concerning this particular point sampling we refer to ourprevious findings in [17, Subsection 3.9].

To evaluate the methods’ approximation behaviour, we consider the resulting maximal errorǫ = ‖s − u‖∞ among the barycenters of the triangles in the stencils’ corresponding Delaunaytriangulation, see Figure 1. Similar to the assessment in our previous paper [32] concerningkernel-based reconstruction for the scalar case, we consider using the vector field

[

u

v

]

=

[

cos(k π (x − 1

4)) sin(k π (y − 1

4))

sin(k π (x − 1

4)) cos(k π (y − 1

4))

]

,




(a) N = 3 (b) N = 9 (c) N = 15 (d) N = 21

Figure 1. Stencils of different sizes N for accuracy tests.

where increasingly denser data sets are used. The data sets were generated by scaling theinterpolation points X, so that qX ≡ h = 2−i, i = 0, 1, 2, 3, 4, for the separation distance

qX = minx,y∈X

x6=y

‖x − y‖

of the interpolation points in stencil X.The numerical results in Table I show the approximation error ǫ(RT0) obtained when using

Raviart-Thomas elements of order zero, RT0, in comparison with the approximation errorǫ(RBF) of Gaussian reconstruction. Table I also shows the corresponding spectral conditionnumber κ(A) of the Gaussian interpolation matrix A. For the purpose of further comparison,corresponding approximation errors obtained with a RT1 reconstruction are also shown, alongwith estimates for the resulting convergence rates. The test were performed for a decreasingsequence of separation distances h = 2−i, i = 0, 1, 2, 3, 4, just before the linear system ofGaussian reconstruction becomes numerically unstable, due to too large spectral conditionnumbers, cf. the last column of Table I.

Table I. Comparison between RT0, RT1 and Gaussian reconstruction for the 15-point stencil inFigure 1 (c). The relative approximation error ǫ, approximate convergence rate, and spectral condition

number κ(A) of the corresponding Gaussian interpolation matrix A are shown, respectively.

h ǫ(RT0) rate ǫ(RT1) rate ǫ(RBF) rate κ(A)

2−0 7.177 · 10−1 - 7.0920 · 10−1 - 7.248 · 10−1 - 3.625 · 104

2−1 3.070 · 10−1 1.225 1.6431 · 10−1 2.109 1.151 · 10−1 2.654 2.511 · 106

2−2 1.349 · 10−1 1.187 4.4320 · 10−2 1.890 1.451 · 10−2 2.988 1.650 · 108

2−3 6.238 · 10−2 1.112 1.1277 · 10−2 1.974 1.773 · 10−3 3.033 1.064 · 1010

2−4 2.991 · 10−2 1.060 2.6904 · 10−3 2.067 2.178 · 10−4 3.025 6.833 · 1011

For the numerical results in Table I, the 15-point stencil of Figure 1 (c) was utilized.As expected, we obtain linear convergence for Raviart-Thomas reconstruction of order zero,whereas Gaussian reconstruction yields third order accuracy, see Table I. Moreover, despite




the small separation distance of up to h = 2−4, Gaussian reconstruction is very robust. Butfor smaller values of h, the corresponding linear system is ill-conditioned, so that it does notmake sense to further evaluate the method’s accuracy.

In a second test case, we compare the approximation quality of Gaussian reconstruction forfour different stencils of sizes N = 3, 9, 15, 21, displayed in Figure 1. Our numerical results arereflected by Table II (for N = 3, 9, 21) and Table I (for N = 15).

Table II. Gaussian reconstruction for stencils of different sizes N , see Figure 1.

h ǫ (N = 3) rate ǫ (N = 9) rate ǫ (N = 21) rate

2−0 7.177 · 10−1 - 7.156 · 10−1 - 7.180 · 10−1 -2−1 3.070 · 10−1 1.225 1.251 · 10−1 2.516 8.767 · 10−2 3.0342−2 1.349 · 10−1 1.187 1.769 · 10−2 2.823 1.181 · 10−2 2.8922−3 6.238 · 10−2 1.112 2.594 · 10−3 2.770 1.819 · 10−3 2.6982−4 2.991 · 10−2 1.060 4.270 · 10−4 2.603 9.958 · 10−5 4.192

Not too surprisingly, the accuracy order is increasing with the size N of the utilized stencils,see Table I and II. Indeed, the stencil with N = 21 interpolation points yields the best accuracyrate among the four stencils, namely slightly above order four, whereas the stencil with N = 3points yields first order accuracy only. Indeed, for N = 3 the errors differ from those of theRT0 reconstruction in the last decimal digits and the empirical convergence rates achieved bythe two approaches are identical. Although no theorical explanation is available for this fact,we think it shows that, for a smooth vector field, if only 3 normal components are providedas input, the contribution of nonlinear terms in the RBF reconstruction is marginal. We canconclude that, in the tests we performed, for any stencil with more than three interpolationpoints, Gaussian reconstruction is superior to RT0 reconstruction. Further numerical resultssupporting this statement were reported in [2].

5. APPLICATION TO FLUID DYNAMICS PROBLEMS

In this section, we discuss applications of the proposed kernel-based Gaussian reconstructionmethod to computational fluid dynamics models. More specifically, we will refer to modelsfor the shallow water equations, using discretization approaches in which the velocity fieldis represented by its normal components with respect to the mesh edges. The shallow waterequations model the two dimensional flow of a thin fluid layer in domains whose characteristiclength in the horizontal is much larger than the fluid depth. The shallow water equations resultfrom the Navier-Stokes equations when the hydrostatic assumption holds and only barotropicand adiabatic motions are considered. Furthermore, a vertical average is performed, so thatonly mean values for the velocities in the horizontal directions are considered, see e.g. [26].




The shallow water equations can be written as

∂h

∂t+ ∇ ·

(

Hv)

= 0, (10)

∂v

∂t+ (v · ∇)v = −fk × v − g∇h. (11)

Here, v denotes the two-dimensional velocity vector, k is the radial unit vector perpendicular tothe plane on which v is defined (or to the local tangent plane, in case of applications in sphericalgeometry), h is the height of the fluid layer above a reference level, H = h−hs is the thickness ofthe fluid layer, hs is the orographic or bathymetric profile, g is the gravitational constant, andf is the Coriolis parameter. Equations (10)-(11) are the starting point for Eulerian-Lagrangiandiscretizations.

Another widely used formulation for applications to large scale atmospheric dynamics is theso called vector invariant form, see e.g. [41], which can be written as

∂v

∂t= −(ζ + f)k × v −∇

(

gh + K)

. (12)

Here, ζ is the component of relative vorticity in the direction of k and K = ‖v‖2/2 denotesthe kinetic energy. This formulation is usually the starting point for the derivation of energy,potential enstrophy and potential vorticity preserving discretizations, see e.g. [1, 31].

Spatial discretizations with staggered arrangements of the discrete variables are popular forthe shallow water equations, since they allow for better representation of the gravity wavepropagation, see e.g. [29]. On unstructured grid, an analog of a staggered discretization isgiven by the zero order Raviart-Thomas elements RT0, see e.g. [28]. Although high order RTelements are also available, the low order ones, RT0 elements, lead more easily to numericalmethods that exhibit important discrete conservation properties, such as discrete mass orvorticity preservation. These properties are important for a number of applications andvarious methods which take advantage of them are discussed in the two following subsections.Our main point is that the accuracy of these models has been limited so far by the firstorder convergence of the RT0 elements. As it will be shown in the following, the proposedkernel-based Gaussian reconstruction can effectively improve these methods, by achieving amore accurate discretization of the nonlinear momentum advection terms, either in Eulerianor in semi-Lagrangian formulations. Although in general this is not sufficient to raise theconvergence order of the overall methods, models employing kernel-based reconstructionsdisplay significantly smaller errors and have in general less numerical dissipation, makingtheir use attractive for a number of applications.

5.1. EULERIAN SHALLOW WATER MODELS

Eulerian discretizations of equations (10)-(12) have been proposed in [5, 6], which preservediscrete approximations of mass, vorticity and potential enstrophy. These properties areimportant for numerical models of general atmospheric circulation, especially for applicationsto climate modelling. The two time level, semi-implicit scheme in these papers used RTreconstruction to compute the nonlinear terms in the discretization of (12). Here, we willcompare results obtained with a three-time level, semi-implicit time discretization, coupled




to the potential enstrophy preserving spatial discretization of [6], using either the Raviart-Thomas algorithm or a kernel-based reconstruction of the velocity field necessary for thesolution of equation (12). For these tests, we employed Gaussian reconstruction by using the

positive definite Gaussian kernel φ(r) = e−αr2

. Moreover, a 9-point stencil was employed,see Figure 1 (b), using the normal components to the edges of the triangle on which theinterpolation is being carried out and to the edges of its nearest neighbours (i.e., of the triangleswhich have common edges with it). We remark that in this case the velocity components arelying on planes tangent to the sphere at the edge points where the velocity component isdefined. In order to apply Gaussian reconstruction, these vector components are interpretedas vectors in three dimensional space and the procedure described in Section 3 is extended inthe natural way to the three dimensional case.

We consider one stationary and two non-stationary test cases for the shallow water equationsbelonging to the set of standard benchmark problems introduced in [41]. First, we study howthe algorithm performs when applied to test case 3 of this test suite, which consists of asteady-state, zonal geostrophic flow with a narrow jet at midlatitudes. For this test case,an analytic solution is available, so that errors can be computed by applying the numericalmethod at different resolutions. The values of the relative error in various norms, as computedat day 2 with different spatial resolutions and with time step ∆t = 1800 s, is displayed inTables III and IV for both Raviart Thomas elements and Gaussian reconstruction, respectively.The convergence rates remain essentially the same, due to the fact that the second orderdiscretization of the geopotential gradient is the same in both tests. However, it can be observedthat the errors (both in the height and velocity fields) have decreased by an amount that rangesbetween 30 % and 50 %.

Table III. Relative errors for nonlinear terms in shallow water test case 3 obtained by using RT0

reconstruction.

Level ℓ2-error, h ℓ2-error, v ℓ∞-error, h ℓ∞-error, v3 7.42e-3 0.25 2.53e-2 0.334 1.94e-3 5.9e-2 8.1e-3 9.1e-25 6.05e-4 1.27e-2 2.9e-3 1.87e-26 2.54e-4 3.19e-3 1.24e-3 4.17e-3

Table IV. Relative errors for nonlinear terms in shallow water test case 3 obtained by Gaussianreconstruction on a 9-point stencil, see Figure 1 (b).

Level ℓ2-error, h ℓ2-error, v ℓ∞-error, h ℓ∞-error, v3 7.27e-3 0.16 2.08e-2 0.174 1.52e-3 3.38e-2 6.74e-3 5.77e-25 4.05e-4 7.7e-3 1.7e-3 1.22e-26 1.45e-4 2.11e-3 4.8e-4 2.89e-3

We have then considered the non-stationary test case 5 of [41], for which the initial datumconsists of a zonal flow impinging on an isolated mountain of conical shape. The imbalance in




the initial datum leads to the development of a wave propagating all around the globe. Thistest is relevant to understand the response of the numerical model to orographic forcing. Plotsof the meridional velocity component at simulation day 5 are shown in Figure 2, as computedusing a timestep ∆t = 900 s on an icosahedral grid at spatial resolution of approximately240 km. We observe that the meridional velocity field obtained by using RT0 finite elements ismuch less regular than that obtained by Gaussian reconstruction, which is consistent with ournumerical results obtained in reference simulations at higher resolution with spectral models.

Finally, we have considered the non-stationary test case 6 of [41], for which the initial datumconsists of a Rossby-Haurwitz wave of wavenumber 4. This type of wave is an analytic solutionfor the barotropic vorticity equation, which can also be used to test shallow water models ona time scale of up to 10-15 days. The relative vorticity field is shown in Figure 3, as computedat day 5 with a timestep of ∆t = 900 s on an icosahedral grid with a spatial resolution ofapproximately 240 km. It can be observed that, when using RT0 reconstruction, the structureof some vorticity extrema is disrupted, while spurious maxima and minima appear close to thepoles. This is in contrast to the more regular field obtained by Gaussian reconstruction, whichis in better agreement with high resolution reference simulations obtained with spectral models.Furthermore, the relative change in total energy for both model runs is displayed in Figure 4. Itcan be observed that total energy loss is reduced by approximately 30 % when using Gaussianreconstruction, thus reducing the numerical diffusion and improving the energy conservationproperties of the model, which conserves potential enstrophy but not energy, as discussed in [6].We remark that, for Eulerian models, the additional computational cost required by Gaussianreconstruction can be significantly reduced. Indeed, it is possible to compute for each gridcell a set of time independent coefficients that yield the velocity vector at the cell center aslinear combination of the velocity components at the points included in the RBF stencil. Forthe model runs described above, it was observed that Gaussian reconstruction increases therequired CPU time by approximately 20 % in comparison to the simpler scheme RT0.

5.2. EULERIAN-LAGRANGIAN SHALLOW WATER MODELS

Numerical methods for the shallow water equations using formulation (10)-(11) have beenproposed in [8, 23], which couple a mass conservative, semi-implicit discretization onunstructured Delaunay meshes to an Eulerian-Lagrangian treatment of momentum advection.The resulting methods are highly efficient due to their rather weak stability restrictions,while mass conservation allows for their practical (and successful) application to a numberof pollutant and sediment transport problems. A key step of the Eulerian-Lagrangian methodis the interpolation at the foot of characteristic lines. In the papers quoted above, this isperformed by RT0 elements or by low order interpolation procedures based on area weightedaveraging. These interpolators have at most first order convergence rate and can introducelarge amounts of numerical diffusion, which limits their applicability especially in long termsimulations.

Firstly, the test proposed in [42] has been carried out, in which a 800 m long and 800 m widebasin was considered. The domain was discretized by an unstructured triangular mesh with26,812 elements and 13,868 nodes, corresponding to a horizontal resolution of approximately2 km. The basin depth was taken to be 20 m. At the inflow boundary, a Dirichlet condition wasimposed on the free surface, given by a sinusoidal wave with an amplitude of 1 m and a periodof approximately 12 h and the initial velocity field was assumed to be zero. The resulting




(a) Reconstruction by RT0 (b) Gaussian reconstruction

Figure 2. Meridional velocity in shallow water test case 5, obtained by (a) RT0 reconstruction (b)Gaussian reconstruction on a 9-point stencil. The contour line spacing is 6 ms−1. Maximum and

minimum values are approximately ±28 ms−1, respectively.

wavefront, computed after approximately 30 h by the Eulerian-Lagrangian method of [23],using a timestep ∆t = 6 s and either RT0 elements or Gaussian reconstruction at the foot of thecharacteristic, is displayed in Figure 5. It can be observed that for Gaussian reconstruction thewavefront is much sharper. The maximum in the free surface elevation (which would be equalto the maximum boundary value in the linear regime) is better captured by approximately10 %. This can lead to significant improvements in a number of relevant applications, such asflooding prediction in the Venice Lagoon, which is at end of a closed sea basin of approximatelythe same magnitude as the simulated channel.

Furthermore, another shallow water test involving a closed rectangular basin, 150 m longand 15 m wide, was performed, whose discretization is given by an unstructured triangularmesh with 3,646 elements and 1,984 nodes. For the free surface, an unbalanced initial datumwas assumed, given by η(x) = h0 cos(xπ/150). The amplitude of the disturbance was takento be equal to h0 = 0.1 and the initial velocity field was assumed to be zero. The resultingfree oscillations have been simulated by the same method described above, using either RT0

elements or Gaussian reconstruction at the foot of the characteristic. The free oscillations ofthe fluid were simulated for a total of 100 s with a timestep ∆t = 0.1 s. The time evolutionof kinetic energy is shown in Figure 6, while the height field values computed throughoutthe simulation in an element close to one of the boundaries are shown in Figure 7. It can beobserved that the energy dissipation caused by the interpolation of the Eulerian-Lagrangianmethod is reduced by 20 % when using Gaussian reconstruction, whereas the maxima andminima in the height field are improved by approximately 10 %.

In other tests, even larger improvements were observed. For example, a square domainof width 20 m was considered, which was discretized by an unstructured triangular meshwith 3,984 elements and 2,073 nodes. A constant basin depth of 2 m was assumed. At initialtime, still water was assumed and the free surface profile was taken to be a Gaussian hill




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centered at the center of the domain, with amplitude 0.1 m and standard deviation 2 m. Inabsence of any explicit dissipative term, the total energy of the system should be conserved.The free oscillations of the fluid were simulated for a total of 6 s with a timestep ∆t = 0.01 s.The time evolution of total energy is shown in Figure 8. It can be observed that the energydissipation caused by the interpolation of the Eulerian-Lagrangian method is reduced by 40 %when using Gaussian reconstruction.

We remark that in the case of Eulerian-Lagrangian models the extra computational costdue to the use of Gaussian reconstruction is higher than in the Eulerian case. This is becausethe coefficients which yield the velocity vector at the cell center as linear combination ofthe velocity components have to be recomputed at each time step for each of the trajectorydeparture points.

6. CONCLUSIONS

The utility of diagonally-scaled radial kernel functions for accurate reconstruction of vectorfields in fluid dynamics problems has been demonstrated. The theory of the utilizedreconstruction method has been reviewed and adapted to applications in computational fluiddynamics. Important computational aspects concerning the implementation of the kernel-based Gaussian reconstruction method were discussed. As an example of the advantages of theproposed techniques, some applications to the discretization of the shallow water equationshave been presented. Many numerical methods for these equations already exist (see e.g. amongmany others [], []), some of which can achieve much higher order convergence and employ morecoherent treatment of all the different terms in the equations. However, we believe that our




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results prove that the use of radial kernel functions to improve the accuracy of low orderRaviart-Thomas elements RT0 yields a convenient compromise between higher order methodsand simpler lower order approaches. In particular, our method allows to retain importantdiscrete conservation properties, that to the best of our knowledge have never been provenfor high order finite element approaches, while maintaining the simplicity of the RT0, thatalso enables the use of very efficient semi-implicit discretizations such as those proposed in [6].Furthermore, semi-Lagrangian techniques such as those proposed in [8, 23] can benefit of theproposed reconstruction technique by effectively reducing the amount of numerical diffusionintroduced by the interpolation step.

ACKNOWLEDGEMENTS

The authors were partly supported by the Max-Planck Institute for Meteorology, Hamburg,through the ICON project. Special thanks go to J. Baudisch, who has provided first preliminaryresults through his Master Thesis [2]. Many useful conversations with W. Sawyer on theapplication of radial kernels in computational fluid dynamics are gratefully acknowledged,as well as earlier contributions by T. Heinze and L. Kornblueh to the development of theICON shallow water model.

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